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Vladimir I. Man'ko

Publications and source records attributed to Vladimir I. Man'ko.

At least 37 records · Page 2Linked to original sources
arXiv2019

Quantum suprematism picture of Malevich's squares triada for spin states and the parametric oscillator evolution in the probability representation of quantum mechanics

Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin--systems with discrete variables. New entropic--information inequalities are obtained for Franck--Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.

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arXiv2019

Correlations in a system of classical--like coins simulating spin-1/2 states in the probability representation of quantum mechanics

An analog of classical "hidden variables" for qubit states is presented. The states of qubit (two-level atom, spin-1/2 particle) are mapped onto the states of three classical--like coins. The bijective map of the states corresponds to the presence of correlations of random classical--like variables associated with the coin positions "up" or "down" and the observables are mapped onto quantum observables described by Hermitian matrices. The connection of the classical--coin statistics with the statistical properties of qubits is found.

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arXiv2018

Geometry and Entanglement of Two-Qubit States in the Quantum Probabilistic Representation

A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres--Horodecki positive partial transpose (ppt)-criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.

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arXiv2018

Probability representation of quantum observable and quantum states

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.

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arXiv2018

Triangle geometry for qutrit states in the probability representation

We express the matrix elements of the density matrix of the qutrit state in terms of probabilities associated with artificial qubit states. We show that the quantum statistics of qubit states and observables is formally equivalent to the statistics of classical systems with three random vector variables and three classical probability distributions obeying special constrains found in this study. The Bloch spheres geometry of qubit states is mapped onto triangle geometry of qubits. We investigate the triada of Malevich's squares describing the qubit states in quantum suprematism picture and the inequalities for the areas of the squares for qutrit (spin-1 system). We expressed quantum channels for qutrit states in terms of a linear transform of the probabilities determining the qutrit-state density matrix.

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arXiv2018

Triangle geometry of the qubit state in the probability representation expressed in terms of the triada of Malevich's Squares

We map the density matrix of the qubit (spin-1/2) state associated with the Bloch sphere and given in the tomographic probability representation onto vertices of a triangle determining Triada of Malevich's squares. The three triangle vertices are located on three sides of another equilateral triangle with the sides equal to $\sqrt 2$. We demonstrate that the triangle vertices are in one-to-one correspondence with the points inside the Bloch sphere and show that the uncertainty relation for the three probabilities of spin projections +1/2 onto three orthogonal directions has the bound determined by the triangle area introduced. This bound is related to the sum of three Malevich's square areas where the squares have sides coinciding with the sides of the triangle. We express any evolution of the qubit state as the motion of the three vertices of the triangle introduced and interpret the gates of qubit states as the semigroup symmetry of the Triada of Malevich's squares. In view of the dynamical semigroup of the qubit-state evolution, we constructed nonlinear representation of the group U(2).

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arXiv2018

God plays coins or superposition principle for classical probabilities in quantum suprematism representation of qubit states

For three given probability distributions describing positions of three classical coins the quantum density matrix of spin-$1/2$ state is constructed and its matrix elements are associated with triada of Malevich's squares. The superposition principle of spin-$1/2$ states is presented in the form of nonlinear addition rule for these classical coin probabilities. We illustrate the formulas by the statement"God does not play dice - God plays coins".

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arXiv2018

Entropic bounds between two thermal equilibrium states

The positivity conditions of the relative entropy between two thermal equilibrium states $\hatρ_1$ and $\hatρ_2$ are used to obtain upper and lower bounds for the subtraction of their entropies, the Helmholtz potential and the Gibbs potential of the two systems. These limits are expressed in terms of the mean values of the Hamiltonians, number operator, and temperature of the different systems. In particular, we discuss these limits for molecules which can be represented in terms of the Franck--Condon coefficients. We emphasize the case where the Hamiltonians belong to the same system at two different times $t$ and $t'$. Finally, these bounds are obtained for a general qubit system and for the harmonic oscillator with a time dependent frequency at two different times.

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arXiv2017

Hidden correlations and entanglement in single-qudit states

We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.

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arXiv2017

Metric on the space of quantum states from relative entropy. Tomographic reconstruction

In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.

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arXiv2016

Discretization of the density matrix as a nonlinear positive map and entanglement

The discretization of the density matrix is proposed as a nonlinear positive map for systems with continuous variables. This procedure is used to calculate the entanglement between two modes through different criteria, such as Tsallis entropy, von Neumann entropy and linear entropy and the logarithmic negativity. As an example, we study the dynamics of entanglement for the two-mode squeezed vacuum state in the parametric amplifier and show good agreement with the analytic results. The loss of information on the system state due to the discretization of the density matrix is also addressed.

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arXiv2016

Evolution and entanglement of Gaussian states in the parametric amplifier

The linear time-dependent constants of motion of the parametric amplifier are obtained and used to determine in the tomographic-probability representation the evolution of a general two-mode Gaussian state. By means of the discretization of the continuous variable density matrix, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier. The obtained results for the nonlocal correlations are compared with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. This qubit portrait procedure is used to establish Bell-type's inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. Other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.

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arXiv2016

Hidden Bell correlations in the four-level atom

We extend the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit, and qudit with j=3/2. We formulate the extended inequality as the inequality valid for an arbitrary Hermitian nonnegative 4x4-matrix with unit trace for both separable and entangled matrices.

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arXiv2015

Hidden Quantum Correlations in Single Qudit Systems

We introduce the notion of hidden quantum correlations. We present the mean values of observables depending on one classical random variable described by the probability distribution in the form of correlation functions of two (three, etc.) random variables described by the corresponding joint probability distributions. We develop analogous constructions for the density matrices of quantum states and quantum observables. We consider examples of four-dimensional Hilbert space corresponding to the "quantum roulette" and "quantum compass."

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arXiv2014

Deformed entropy and information relations for composite and noncomposite systems

The notion of conditional entropy is extended to noncomposite systems. The q-deformed entropic inequalities, which usually are associated with correlations of the subsystem degrees of freedom in bipartite systems, are found for the noncomposite systems. New entropic inequalities for quantum tomograms of qudit states including the single qudit states are obtained. The Araki{Lieb inequality is found for systems without subsystems.

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arXiv2014

Entropy-energy inequalities for qudit states

We establish a procedure to find the extremal density matrices for any finite Hamiltonian of a qudit system. These extremal density matrices provide an approximate description of the energy spectra of the Hamiltonian. In the case of restricting the extremal density matrices by pure states, we show that the energy spectra of the Hamiltonian is recovered for $d=2$ and $3$. We conjecture that by means of this approach the energy spectra can be recovered for the Hamiltonian of an arbitrary finite qudit system. For a given qudit system Hamiltonian, we find new inequalities connecting the mean value of the Hamiltonian and the entropy of an arbitrary state. We demonstrate that these inequalities take place for both the considered extremal density matrices and generic ones.

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arXiv2014

Entanglement and other quantum correlations of a single qudit state as a resource for quantum technologies

The approach to extend the notion of entanglement for characterizing the properties of quantum correlations in the state of a single qudit is presented. New information and entropic inequalities, such as the subadditivity condition, strong subadditivity condition, and monotonicity of relative entropy for a single qudit corresponding to an arbitrary spin state with spin j are discussed. The idea to employ quantum correlations in the single-qudit state, such as the entanglement, for developing a new quantum technique in quantum computing and quantum communication is proposed. Examples of qutrit and qudit with j = 3/2 are considered.

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